The Initial Boundary Value Problem for the Flow of a Barotropic Viscous Fluid, Global in Time

Fiszdon, W.; Zaja̧czkowski, Wojciech M.; Gilbert, Robert P.

doi:10.1080/00036818308839441

Cited by 5 publications

(6 citation statements)

References 7 publications

(5 reference statements)

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“…Hence one utilizes Eq. (2.13) x restricted on S l5 and by proceeding as in Fiszdon and Zajaczkowski [4][5][6], after some long but straightforward calculations one gets -J ύ.ή(σ 2…”

Section: Simentioning

confidence: 99%

“…(Actually, it is enough to assume that these coefficients are C 2 functions of ρ and θ, see Appendix in Sect. 6. )…”

Section: Introductionmentioning

confidence: 98%

“…Finally, the following compatibility conditions have to be assumed In the last thirty years this problem has been intensively studied, firstly investigating uniqueness (see Graffi [8], Serrin [26], Valli [29]) and later (local in time) existence (see Nash [20], Itaya [9], VoΓpert and Hudjaev [32] for the Cauchy problem in R 3 ; Solonnikov [27], Tani [28], Valli [30] for the Cauchy-Dirichlet problem in a general domain with vanishing velocity on the boundary; Fiszdon and Zajaczkowski [4,5] for the Cauchy-Dirichlet problem in a bounded domain too but in the case of an inflow only, i.e., S x = dΩ; Lukaszewicz [12,13] for the Cauchy-Dirichlet problem in a bounded domain in the case of inflow and outflow, i.e. dΩ = S 1 uS 2 , or Si=0).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, a global existence result has been proved by Matsumura and Nishida in IR 3 [15,16]. Afterwords, Matsumura and Nishida [17,18], Valli [31] have shown global existence in a bounded domain Ω with U\ dΩ = 0, and Fiszdon and Zajaczkowski [4,6] in the case of an inflow only. (However, in [4,6,31] only the case of a barotropic fluid (i.e., p -p(ρ)) is considered.…”

Section: Introductionmentioning

confidence: 99%

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Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

1986

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We consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary. By means of some a priori estimates we prove the existence of a global (in time) solution. Moreover, as a consequence of a stability result, we show that there exist a periodic solution and a stationary solution.Partially supported by G.NAFA of C.N.R. (Italy)

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“…Hence one utilizes Eq. (2.13) x restricted on S l5 and by proceeding as in Fiszdon and Zajaczkowski [4][5][6], after some long but straightforward calculations one gets -J ύ.ή(σ 2…”

Section: Simentioning

confidence: 99%

“…(Actually, it is enough to assume that these coefficients are C 2 functions of ρ and θ, see Appendix in Sect. 6. )…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

1986

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“…The local existence and uniqueness of smooth solutions were proved by Serrin [34] and Nash [31]. The local existence of strong solutions with Sobolev regularity was constructed by Solonnikov [36], Valli [37] and Fiszdon-Zajaczkowski [12]. Matsumura and Nishida [29,30] established the first global strong solutions for small perturbations of a linearly stable constant state ( ∞ , 0) in three dimensions.…”

Section: Introductionmentioning

confidence: 99%

A Low-Frequency Assumption for Optimal Time-Decay Estimates to the Compressible Navier–Stokes Equations

2019

Commun. Math. Phys.

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The global existence issue in critical spaces for compressible Navier-Stokes equations, was addressed by Danchin (2000) and then developed by Charve-Danchin (2010), Chen-Miao-Zhang (2010) and Hasport (2011) in more general L p setting. Our aim is to exhibit (more precisely) time-decay estimates of solutions constructed in the critical framework. As we known, the low-frequency assumption usually plays a key role in the large-time asymptotics of solutions, which was firstly observed in usual L 1 space by Matsumura and Nishida [30]. In this paper, we claim a new low-frequency assumption for barotropic compressible Navier-Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov spaceḂ −σ 1 2,∞ with the regularity σ1 ∈ (1−d/2, 2d/p−d/2] , then a sharp time-weighted inequality including enough time-decay information can be available, where optimal decay exponents for the high frequencies are also exhibited. The proof mainly depends on tricky and non classical Besov estimates. As a by-product, those optimal time-decay rates of L q-L r type are well captured in the critical regularity framework.

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The global existence of solutions to the equations of motion of a viscous gas with an artificial viscosity

Neustupa

1991

Math Methods in App Sciences

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Communicated by E. MeisterWe deal with the system of equations of motion of a viscous barotropic fluid. The system contains an artificial viscosity, which depends on the density p of the fluid and is identically equal to zero for p E (0, p 2 ) (where p2 is a given positive number). If p2 is chosen sufficiently large, the system coincides with the Navier-Stokes equations and the equation ofcontinuity if the density has values that actually appear in real flows. The velocity is assumed to be equal to zero on the boundary of the flow field and the body force is not taken into account. Initial conditions need not be 'small enough'. Applying the method of discretization in time, the existence of a weak solution on an interval of an arbitrary (but finite) length is proved and an estimate of the energy character is derived.

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The Initial Boundary Value Problem for the Flow of a Barotropic Viscous Fluid, Global in Time

Cited by 5 publications

References 7 publications

Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

A Low-Frequency Assumption for Optimal Time-Decay Estimates to the Compressible Navier–Stokes Equations

The global existence of solutions to the equations of motion of a viscous gas with an artificial viscosity

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